Calendar
Printer-friendly versionThursday, April 9, 2026
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| 4:00pm |
04/09/2026 - 4:00pm The space of immersions of a closed manifold M into Euclidean space is among the most important infinite dimensional manifolds. The most natural Riemannian metrics on this space are reparametrization-invariant Sobolev metrics; these form a hierarchy of metrics, based on their order—the number of derivatives they “see”. They arise as natural extensions of right-invariant metrics on diffeomorphism groups, central to Arnold’s geometric formulation of hydrodynamical equations. Beyond their theoretical significance, they play a crucial role in mathematical shape analysis and geometric data science, where they enable meaningful and robust comparisons between shapes modeled as curves or surfaces.
In 2013, David Mumford conjectured that for orders larger than dim(M)/2 + 1, these geometries are complete. Note, that by the seminal work of Ebin and Marsden a similar statement is known to be true for diffeomorphism groups. In the context of immersions this conjecture was shown to be true in the case of immersed curves in the work of Bruveris, Michor and Mumford. In this talk I will present the first construction of complete metrics on immersions of two-dimensional surfaces, discuss the context and techniques, as well as possible extensions to higher dimensions. Based on joint work with Cy Maor and Benedikt Wirth.
Zoom Link: https://yale.zoom.us/j/95804422721
Location:
Zoom
04/09/2026 - 4:30pm I will describe recent progress towards understanding the gravitational path integral in AdS_3 quantum gravity and its boundary interpretation. A central question is: which spacetime topologies should be included in the path integral, and why? To address this question, we formulate a “statistical bootstrap” that constrains the universal statistics of CFT data in the boundary theory, imposing crossing symmetry and “typicality” (a generalization of the eigenstate thermalization hypothesis). These constraints are geometrized by iterative surgery moves on bulk manifolds that we refer to as the “gravitational machine,” leading to an infinite set of non-handlebody topologies that we argue must be included in the path integral. The machine generates only on-shell (hyperbolic) 3-manifolds, whose partition functions can be computed exactly using Virasoro TQFT. But not all hyperbolic manifolds are produced by this procedure. This reveals a large landscape of consistent sums over topologies. Based on joint work with Alexandre Belin, Lorenz Eberhardt, Diego Liska, and Boris Post. Location:
KT 801
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